Remark 4.2 The conclusion of Proposition REF  also holds if we replace the Cesàro average \(\frac{1}{n}\sum _{k=1}^n \mu ^{k}\ast \delta _x \)  by the sequence of empirical measures. More precisely, for every \(x \in X_\mathcal {C}\) , for \(\mu ^{\mathbb {N}}\) -a.e.  \(a \in H^{\mathbb {N}}\) , the sequence \(\frac{1}{n}\sum _{k=0}^{n-1} \delta _{a_k\ldots a_0 x}\)  converges to a product measure of the same form as in Theorem REF . This follows in the same way, using in addition Breiman's law of large numbers (see e.g.  [1]}) and the corresponding empirical measure equidistribution results of Benoist–Quint.

